C₂×C₃⋊#19128624 #44544875

order	32004
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₃⋊#19128624
unambiguous semidirect product	(C₂×C₃)⋊#19128624

Splittings

Fitting, socle	C₂×C₃×C₇×C₁₂₇ ⋊ C₂×C₃
derived	C₃×C₇×C₁₂₇ ⋊ C₂²×C₃
Hall	C₃×#14581888 ⋊ C₂²
Hall	C₇×C₁₂₇ ⋊ C₂×C₃×D₃
Sylow	C₁₂₇ ⋊ C₂×D₃×ΓL(1,𝔽₈)
Sylow	C₇ ⋊ C₂×C₃⋊(D₁₂₇⋊C₃)

Subgroups

	subgroup	G/H
center, hypercenter	C₂	C₃⋊#19128624
commutator, cosocle, hypocenter	C₃×C₇×C₁₂₇	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₇×C₁₂₇	C₂×C₃
LCS	G ▷ C₃×C₇×C₁₂₇
UCS	C₁ ◁ C₂

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 139
transitive degree	μ_t(G) = 5334

A minimal faithful permutation representation — 2 generators on 139 points:

(4,5)(7,8,9)(10,12,11)(14,18,19,15,16,20)(17,26,27,21,22,36)(23,39,40,29,30,53)(24,38,34,32,28,25)(31,58,59,42,43,76)(33,56,57,45,41,50)(35,37,55,47,48,51)(44,81,82,61,62,104)(46,79,80,64,60,73)(49,54,78,66,67,74)(52,93,91,70,71,69)(63,109,110,84,85,130)(65,107,108,86,83,101)(68,77,106,88,89,102)(72,117,118,94,95,97)(75,123,115,98,99,92)(87,132,133,112,111,128)(90,105,131,114,113,129)(96,135,136,119,120,126)(100,139,138,124,125,122)(103,137,134,121,127,116);(1,2)(3,4,5)(6,7,8,10,9,11,12)(13,14,17,23,31,44,63,85,111,120,125,127,129,130,114,88,66,47,34,19,18,24,33,46,65,87,113,62,83,95,99,102,104,128,134,115,91,57,27,38,26,37,52,72,96,121,89,43,60,71,74,76,101,126,138,118,80,40,55,56,39,54,75,100,124,98,67,30,41,51,53,73,97,122,136,108,59,78,93,79,58,77,103,119,94,70,48,22,28,36,50,69,92,116,133,82,106,123,117,107,81,105,112,86,64,45,32,16,20,25,35,49,68,90,110,131,137,139,135,132,109,84,61,42,29,21,15)

Conjugacy classes & character table

912 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32004, 257);

C2×C3⋊#19128624 #44544875